Model-Based Diagnosis of Hybrid Systems

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Model-Based Diagnosis of Hybrid Systems

• Papers by
• Sriram Narasimhan and
• Gautam Biswas
• Presented by John Ramirez

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Introduction

• Modern systems are complex, and include
  supervisory control that switches modes of
  behavior.
• The controller is a software program and is not
  tightly meshed with the continuous plant dynamics.

Plant
Actuators
Sensors
Sensor values
Discrete Signals
Supervisory controller
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Introduction

• The continuous dynamics of the plant are defined
  by differential and algebraic equations.

q(t) is the discrete model
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Fault Detection and Isolation (FDI)

• The goal of this presentation is to briefly
  overview the study of FDI in hybrid systems with
  supervisory controllers.
• System faults may be component, actuator, sensor,
  and controller faults. (We do not deal with the
  later)
• The methodology we will cover combines
  qualitative and quantitative reasoning techniques
  to perform parameterized fault isolation of plant
  component faults.

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Modeling for Diagnosis

• Controller Model
• Plant Model

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Modeling for Diagnosis

• Controller Model
• The primary model of the controller is
  implemented as a finite state machine (FSM).
• States of the FSM correspond to the states of the
  controller, which in turn define modes of the
  physical plant(q(t)).
• The Transitions determine the conditions for
  switching states.

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Modeling for DiagnosisController Model
t2
t10
t1
t4
t5
t3
t7
t11
t9
t6
t8
Controller Model for 3 tank system
Flow source 1
Flow source 2
Valve
Three Tank system
C capacitance
R3
R5
R resistance
Tank 2 (C2)
Tank 2 (C2)
Tank 1 (C1)
R2
R4
R6
R1
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Modeling for Diagnosis

• Plant Model
• Hybrid Bond Graph Models (HBG).
• State equations and temporal causal graph (TCG)
  can be systematically derived from the bond graph
  representation of the system.
• State equations along with the TCG constitute our
  diagnosis models.

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Methodology for Hybrid Diagnosis

• Hybrid observer follows the continuous dynamics
  of the plant and identifies discrete mode
  changes.
• Fault detection mechanism signals a fault when
  the observer cannot compensate for differences
  between observed and expected behavior.
• Fault isolation mechanism generates candidate
  faults and refines them with the hybrid model and
  measurement from the system.

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Methodology for Hybrid Diagnosis

• The following information is assumed to be
  available to all modules
• HBG
• FSA
• FSM
• ? A all possible autonomous events in the
  system
• U inputs
• Y system outputs
• Parameters nominal

u
y
System
r
Observer and mode detector
Hybrid models
Diagnosis models
Fault isolation
Fault detection
Fault Hypotheses
Diagnosis System Architecture
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Methodology for Hybrid DiagnosisAlgorithm
1Diagnosis Module

• MODULE DIAGNOSE(Minitial,Xinitial)
• // Observe the system until a fault is detected
• ltStackM, YestimatedgtOBSERVER(Minitial,Xinitial)
• //Convert the quantitative residuals to
  qualitative values
• QualResidualcurrent SIGNAL_TO_SYMBOL(Y,Yestimate
  d)
• //Back propagate across modes to identify fault
  candidates
• BackHorizon2
• ListcandidatesHYBRID_BACK_PROP(StackM,QualResidua
  lcurrent,BackHorizon)
• //Forward propagete across modes to isolate the
  fault
• ListcandidatesHYBRID_FAULT_OBSERVER(Listcandidate
  s,Yestimated)
• END DIAGNOSE

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Hybrid Diagnosis Problem
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Fault IsolationBackground

• The type of plant model employed determines the
  scheme to be employed.
• Traditional schemes for the continuous domain use
  structured and directional residual approaches.
• Extending these continuous methodologies to
  hybrid systems becomes intractable.

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Fault Isolation

• The approach we will follow involves hypotheses
  generation and hypotheses refinement.
• Qualitative approach for hypotheses generation.
• Qualitative-quantitative combined approach for
  hypotheses refinement.

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Fault IsolationHypotheses Generation

• For initial hypotheses generation we have to back
  propagate across modes.
• The assumption that the controller model is
  correct implies that the observer predicted the
  correct mode sequence till the fault occurred.
  Therefore, the mode in which the fault occurred
  must be in the predicted trajectory of the
  observer.

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Hypotheses GenerationTCG generation

• Effort and flow variables are vertices
• Relation between variables as directed edges
• implies that two variables associated with the
  edge take on equal values, 1 implies direct
  proportionality,-1 implies inverse
  proportionality.
• Edge associated with component represents the
  components constituent relation.

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Hypotheses GenerationAlgorithm 2Hybrid Back
Propagation

• MODULE HYBRID_BACK_PROP(StackM, QualRi,
  BackHorizon)
• //Generate candidates in each mode in the mode
  trajectory.
• ltMcurrent, TimecurrentgtPop(StackM)
• TCGcurrentGET_TCG(HBG, Mcurrent)
• //Back propagate in selected mode for candidates
  in the mode
• FcurrentCONTINUOUS_BACK_PROP(TCGcurrent,QualRi)
  
• Add(Listcandidates,ltMcurrent,Timecurrent,Fcurren
  tgt)
• Count0
• //Go back in the mode horizon upto BackHorizon
  number of nodes
• While(CountltBackHorizon)
• //Select next mode in mode trajectory and
  calculate TCG
• ltMnext, TimenextgtPop(StackM)
• TCGnext, GET_TCG(HBG, Mnext)
• // Propagate qualitative deviations across modes
• QualRnextBACK_PROP_ACROSS_MODES(Mcurrent,
  Mnext, QualRi)
• //Back propagate in selected mode for candidates
  in the mode
• FnextCONTINUOUS_BACK_PROP(TCGnext,
  QualRnext)
• Add(Listcandidates,ltMnext,Timenext,Fnext,1gt)
• End While

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Roll Back Process

• Qualitative Hypotheses Generation
• Back propagate through TCG in current mode to
  identify candidates
• Back propagate across mode transitions using
  transition conditions (need to account for reset
  conditions, and change in plant configuration
  invert qualitatively)
• Repeat same process for previous modes to
  identify more candidates

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Fault IsolationHypotheses Refinement

• First apply a qualitative forward propagation for
  each hypothesized fault candidate.
• To take into account mode changes, all possible
  modes changes from the current mode are
  hypothesized.
• A candidate is dropped when the predictions do
  not match the observations across all of the
  hypothesized modes
• Apply a quantitative parameter estimation on
  remaining candidates.
• This approach works within a single continuous
  mode.

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Hybrid Diagnosis Problem
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Quick Roll Forward

• Goal Get to current mode, so parameter
  estimation can be applied to refine faults and
  identify fault magnitude
• Lemma 2 Sequence of k mode transitions in any
  order drives the system to the same final model
• Requires tracking of transients by progressive
  monitoring in continuous regions of space. Taylor
  series expansion defines qualitative fault
  signatures. Residual r(t) after fault can be
  described as
• Progressive Monitoring Match qualitative
  magnitude and slope of measurement signal
  transient against fault signature

Fault signature qualitative form of
derivatives Qualitative form of
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Quick Roll Forward

• In continuous case, mismatch implies fault
  hypothesis is not consistent. However, in hybrid
  tracking, it may imply that we are not in the
  right mode. We need to identify the current mode
  (roll forward)
• All controlled transitions are known, but we
  have to hypothesize autonomous transitions since
  observer can no longer predict them correctly
• Use fault signatures to hypothesize mode
  transitions

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Parameter Estimation (Real Time)

• Derive transfer function model in current mode
  with only one unknown (fault parameter)
• Initiate fault observer filter for each fault
  hypothesis
• least squares estimator for parameter estimation
• Test for convergence identifies true fault
  candidate

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Least Square Estimation from IOE
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Parameter Estimation Example
Plot of prediction error
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Quantitative Parameter Estimation Issues

• Deriving the simplified one unknown parameter
  equation for least square estimator
• Convergence to local minima need good initial
  estimates
• Need for persistent excitation in input
  mitigated to some extent by reducing it to a one
  parameter estimation problem
• Measurement noise leads to biased estimates
  need to apply more sophisticated techniques IVM
  methods

Observation What is good for qualitative FDI is
not always good for quantitative identification
using least squares methods
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Summary

• Model for Diagnosis
• Controller Model
• FSM
• Plant Model
• HBG
• Fault Isolation
• Hypotheses Generation
• TCG
• Hypotheses Refinement
• Parameter Estimation

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Conclusion

• By having the supervisory controller model and
  assuming that our model is correct, we do not
  have to make the assumption that faults are
  detected in the mode in which they occur, and we
  still are able to avoid the intractability
  problem.
• Combination of qualitative quantitative
  approaches suitable for online diagnosis
• Approach different from discrete-event approaches
  of Lunze and Sampath